# Logic and Set Theory/Implications, equivalences

## Implications

We begin by introducing a fundamental relation in logic, the implication.

To do so, we first define what a logical statement is. A logical statement ${\displaystyle P(x_{1},\ldots ,x_{n})}$ dependent on several variables, which we denote by ${\displaystyle x_{1},\ldots ,x_{n}}$, is a statement, which, dependent on what values ${\displaystyle x_{1},\ldots ,x_{n}}$ take, is either true or false. For example, consider the statement

${\displaystyle P(x_{1},x_{2}):=(x_{1}\leq x_{2})}$.

Then we will have that ${\displaystyle P(1,2)}$ is true, and ${\displaystyle P(2,1)}$ is false; indeed, ${\displaystyle 1\leq 2}$, but certainly not ${\displaystyle 2\leq 1}$.

With this in mind, we may define the implication.

Definition (implication):

Let ${\displaystyle P(x_{1},\ldots ,x_{n})}$, ${\displaystyle Q(x_{1},\ldots ,x_{n})}$ be logical statements dependent on ${\displaystyle x_{1},\ldots ,x_{n}}$. Saying that ${\displaystyle P}$ implies ${\displaystyle Q}$ (written ${\displaystyle P\Rightarrow Q}$) means that whenever ${\displaystyle P(x_{1},\ldots ,x_{n})}$ is true for given ${\displaystyle x_{1},\ldots ,x_{n}}$, then ${\displaystyle Q(x_{1},\ldots ,x_{n})}$ is also true.